Godel's ACTUAL Proof ☺
- (∃x)Dem(x, z) means there is a proof with Gödel Number x which demonstrates z or more simply z is demonstrable
- ~ (∃x) Dem(x, z) means z is not demonstrable
- A specific case of this expression is obtained by swapping z by Sub(y, 17, y)
- We get ~ (∃x) Dem(x, Sub(y, 17, y)) - lets call this formula B
- B means: the formula with Gödel number Sub(y, 17, y) is not demonstrable
- Lets call the Gödel number of B: n
- Lets replace all instances of the variable y with n to get ~ (∃x) Dem(x, Sub(n, 17, n)) – lets call this formula G with a
Gödel number of g - G means: the formula with Gödel number Sub(n, 17, n) is not demonstrable
- But Sub(n, 17, n) is what we did on line 7 therefore Sub(n, 17, n) has a Gödel number of g
- Therefore, G means: the formula with Gödel number g is not demonstrable
- Gödel proved that G is demonstrable if and only if ~G is demonstrable
- But if the calculus is to be consistent, then G can never be demonstrable otherwise there would be inconsistencies as
its formal negation must also be demonstrable - Since G and its negation ~G are tautological (they are the only 2 options so one must be true) either G is true or ~G is
true. - It is easy to see that G is true as it states that ‘G is not demonstrable’ which we have just proved to be the case if the
calculus is to be consistent - Therefore G is true and can not be demonstrated. Therefore the calculus is incomplete
